Exercises in Classical Ring Theory pp 35-68 | Cite as

# Jacobson Radical Theory

Chapter

## Abstract

The Jacobson radical of a ring *R*, denoted by rad *R*, is the intersection of the maximal left ideals of *R*. This notion is left-right symmetric; in particular, rad *R* is an ideal of *R*. A good way to understand rad *R* is to think of it as the ideal of elements annihilating all left (resp. right) simple *R*-modules. The Jacobson radical is also closely tied in with *U*(*R*), the group of units of *R*. In fact, rad *R* is the largest ideal *R* such that 1 + *R* ⊆ *U*(*R*).

## Keywords

Prime Ideal Maximal Ideal Direct Summand Left Ideal Group Ring
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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## Copyright information

© Springer Science+Business Media New York 1995